[Paper Review] The Gauged Unparticle Action

Alexis Licht|ArXiv.org|Jun 22, 2008

Particle physics theoretical and experimental studies4 references3 citations

TL;DR

This paper demonstrates the equivalence between two formulations of the gauge-invariant unparticle action: one using an open Wilson line to ensure gauge invariance, and another using an integral-differential operator formalism that avoids the Wilson line. The authors show that both approaches yield identical Feynman vertices for unparticle-gauge boson interactions, and they develop a diagrammatic technique to efficiently compute these vertices using the operator formalism, providing a practical method for higher-order calculations in unparticle physics with gauge symmetry.

ABSTRACT

We show that the unparticle action that is made gauge invariant by the inclusion of an open Wilson line factor can be transformed into the integral-differential operator action that avoids the use of the Wilson line factor. The two forms of the action should therefore give the same Feynman diagrams. We also show that it is relatively easy to construct Feynman diagrams using the operator action.

Motivation & Objective

• To resolve inconsistencies in the Mandelstam condition used in Wilson line-based gauging of unparticle actions.
• To establish equivalence between the Wilson line formalism and the integral-differential operator formalism for gauge-invariant unparticle actions.
• To develop a diagrammatic technique for computing unparticle-gauge boson vertices using the operator formalism.
• To provide a practical method for computing higher-order unparticle-gauge interactions without relying on path-ordered Wilson lines.

Proposed method

• Uses a Hilbert space formalism with abstract position and momentum operators to represent unparticle and gauge fields as kets.
• Introduces the operator $ D_{\mu} = P_{\mu} + gA_{\mu} $, which transforms covariantly under gauge transformations, enabling gauge-invariant actions.
• Employs the branch integral formula $ z^n = -\frac{e^{\pi i n}}{\pi} \sin(\pi n) \int_0^\infty dx \frac{x^n}{x - z} $ to express the unparticle propagator in terms of an integral over mass $ M^2 $.
• Derives the gauge-invariant action as $ I = K_0 \int_0^\infty dM^2 (M^2)^{2-d_u} \left\langle \Phi_u \right| \frac{1}{M^2 - D^2 - i\varepsilon} \left| \Phi_u \right\rangle $, avoiding the Wilson line.
• Constructs Feynman rules by identifying propagators $ G_M(p) = \frac{1}{M^2 - p^2 - i\varepsilon} $ and vertices from the operator structure.
• Develops a diagrammatic technique where heavy lines represent the $ M^2 $ integral, and dashed lines represent gauge bosons, with vertices derived from $ D^2 $ expansion.

Experimental results

Research questions

• RQ1 Can the Wilson line-based unparticle action be transformed into an equivalent operator formalism without path-ordered factors?
• RQ2 Are the unparticle-gauge boson vertices derived from the Wilson line and operator formalisms identical?
• RQ3 Does the operator formalism allow for a simpler and more systematic computation of unparticle-gauge interactions?
• RQ4 What is the structure of the double gaugeon-unparticle vertex, and how can it be diagrammatically represented?

Key findings

• The Wilson line-based unparticle action and the integral-differential operator formalism are mathematically equivalent, yielding identical Feynman vertices for unparticle-gauge boson interactions.
• The operator formalism avoids the problematic Mandelstam condition on the Wilson line derivative, resolving a known inconsistency in the literature.
• The single gaugeon-unparticle vertex is derived as a sum of three terms involving $ S^{-1}(p) $, $ S^{-1}(p+q) $, and $ S^{-1}(p') $, with momentum-dependent tensor structures.
• The double gaugeon-unparticle vertex arises from three contributions: $ A^2 $, $ \{P,A\}G_M\{P,A\} $, and a self-energy-like diagram, with the latter derived via diagrammatic rules.
• The diagrammatic technique allows direct reading of vertex structures from the operator action, significantly simplifying the computation of higher-order unparticle amplitudes.
• The vertex expressions are explicitly given in momentum space, with the full amplitude for the double vertex expressed in terms of $ G_M(p) $, $ S^{-1}(p) $, and $ T^a $ group generators.